Maharashtra BoardEnglish MediumSTD 11 ScienceMathsContinuity1 Mark
Question
Show that there is a root for the equation $x^3 – 3x = 0$ between $1$ and $2$.
✓
Answer
Let $f(x) = x^3 – 3x$
$f(x)$ is a polynomial function and hence it is continuous for all $x ∈ R$.
A root of $f(x)$ exists, if $f(x) = 0$ for at least one value of $x$.
$f(1) = (1)^3 – 3(1) = -2 < 0$
$f(2) = (2)^3 – 3(2) = 2 > 0$
$\therefore f(1) < 0$ and $f(2) > 0$
$\therefore $ By intermediate value theorem,
there has to be point $‘c’$ between $1$ and $2$ such that $f(c) = 0$.
There is a root of the given equation between $1$ and $2$.
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